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Results shown of the identification of a three input two output pilot plant evaporative crystallizer (970 litre), in which the Crystal Size Distribution (CSD) is the main characteric of the process to be modelled, and subsequently to be controlled. The experimental situation is characterized by short observaron times and a bad signal to noise ratio of the CSD measurement device. A three step identification procedure is applied, consisting of high order ARX modelling reduction though approximate realization based on step response matrices, and finally output error optimization in a pseudo-canonical (overlaping) state space parametrization.

In prediction error identification, confidence regions are most commonly derived from the asymptotic statistical properties of the parameter estimator. Therefore, these confidence regions are only asymptotically valid and, for finite samples, their actual coverage rate can be smaller than the desired coverage rate. In this paper, we analyze the influence of the SNR and of the type of model structure on the difference between the actual and desired coverage rates. In addition, we propose alternatives to the classical approach to constructing probabilistic confidence regions for Box-Jenkins systems.

Commonly, controllers for Linear Parameter- Varying (LPV) systems are designed in continuous-time using a Linear Fractional Representation (LFR) of the plant. However, the resulting controllers are implemented on digital hardware. Furthermore, discrete-time LPV synthesis approaches require a discrete-time model of the plant which is often derived from continuous-time first-principle models. Existing discretization approaches for LFRs suffer from disadvantages like alternation of dynamics, complexity, etc. To overcome the disadvantages, novel discretization methods are derived. These approaches are compared to existing techniques and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling.

High control performance is essential in many precision positioning applications, such as control of the vertical sample position in a atomic-force microscope (AFM). This paper investigates the impact of load flexibility on piezoelectrically actuated positioning systems in terms of control performance. The modeling method used combines modal analysis with simple transfer function manipulations, and shows, how the load dynamics may influence the control performance. The analysis is experimentally verified on a commercial AFM system.

A common approach to fluid flow modelling is based on a fine spatial discretization of the Navier-Stokes equations, which results in a large number of flow variables (i.e., microstates). For the purpose of flow control, the flow variables may be considered to be aggregated on a macroscopic level that goes beyond the fine modelling grid for a practical application. This paper addresses the problem of stabilizing fluid motions using a macroscopic variable that describes a flow pattern on a macroscopic level of observation. The results are illustrated using a lid driven cavity case in order to verify applicability of the macroscopic quantitative observation for a feedback control design.

A discrete-time linear parameter-varying (LPV) model can be seen as the combination of local LTI models together with a scheduling signal dependent function set, that selects one of the models to describe the continuation of the signal trajectories at every time instant. An identification strategy of LPV models is proposed that consists of the separate approximation of the local model set and the scheduling functions. The local model set is represented as a linear combination (series expansion) of orthonormal basis functions (OBFs). The expansion coefficients are dynamically dependent (weighting) functions of the scheduling parameters (depending on time shifted scheduling). To approximate this dependency class with a static one (non-shifted scheduling), a feedback-based structure of the weighting functions is introduced. The proposed model structure is identified in a two step procedure. First the OBFs, that guarantee the least asymptotic worst-case modeling error for the local models, are selected through the fuzzy Kolmogorov c-Max approach. With the resulting OBFs, the weighting functions are identified through a separable least-squares algorithm. The method is demonstrated by means of simulation examples and analyzed in terms of applicability, convergence, and consistency of the model estimates.

In this paper we propose a virtual closed loop model parameterization to perform system identification. This parameterization is designed to achieve specific goals. We show that the method includes, as special cases, known methods for closed loop identification and also offers additional flexibility. We analyze the ramifications of the new tailor-made parameterization for systems operating in closed loop. The approach exploits a property of Box-Jenkins models in order to minimize the bias arising from feedback and noise model mismatch.

Linear time-invariant, finite dimensional discrete time systems very often are specified in a polynomial form representation in either a forward or backward shift operator (sometimes called MFD and ARMA form). Considering dynamical systems in terms of their behaviour, i.e. the set of admissible signal trajectories, it is shown that there exists a unifying theory for the determination of McMillan degree and Kronecker indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases. Moreover the notions of McMillan degree and Kronecker indices can be generalized to noncontrollable and noncausal systems.

A global and a local identification approach are developed for approximation of linear parameter-varying (LPV) systems. The utilized model structure is a linear combination of globally fixed (scheduling-independent) orthonormal basis functions (OBFs) with scheduling-parameter dependent weights. Whether the weighting is applied on the input or on the output side of the OBFs, the resulting models have different modeling capabilities. The local identification approach of these structures is based on the interpolation of locally identified LTI models on the scheduling domain where the local models are composed from a fixed set of OBFs. The global approach utilizes a priori chosen functional dependence of the parameter-varying weighting of a fixed set of OBFs to deliver global model estimation from measured I/O data. Selection of the OBFs that guarantee the least worst-case modeling error for the local behaviors in an asymptotic sense, is accomplished through the fuzzy Kolmogorov c-max approach. The proposed methods are analyzed in terms of applicability and consistency of the estimates.

In prediction error identification model uncertainty bounds are generally derived from the statistical properties of the parameter estimator. These statistical properties reflect the variability in the estimated model under repetition of experiments with different realizations of the measured signals. However when the primal interest of the identification is in quantifying the uncertainty in an estimated parameter on the basis of one single experiment, this is not necessarily the best and only approach. In the alternative paradigm that is presented here, not the covariance of the estimator will be used for bounding the model uncertainty, but an a posteriori bound on the error in the estimated parameter will be constructed that is structurally dependent on the particular data sequence. This will allow simpler computations for probabilistic model uncertainty bounds also applicable to the situation of approximate modelling (S ∉ M) and to model structures that are nonlinear in the parameters, such as Output Error (OE) models.

Classical system identification methods result in an identified parametric plant and noise model. Due to modelling errors, the estimated variance of the plant model can fail completely to give a reasonable idea of the amplitude of the model errors while this is not detected at all in the validation test. A robust method is presented to validate the parametric variance estimate, and to indicate the reliable frequency bands where the estimated variance can be safely used as an indication of the remaining model errors. Outside these bands the model is unreliable (neither validated or invalidated).

A fuzzy clustering approach is developed to select pole locations for Orthonormal Basis Functions (OBFs), used for identification of Linear Parameter Varying (LPV) systems. The identification approach is based on interpolation of locally identified Linear Time Invariant (LTI) models, using globally fixed OBFs. Selection of the optimal OBF structure, that guarantees the least worst-case local modelling error in an asymptotic sense, is accomplished through the fusion of the Kolmogorov n-width (KnW) theory and Fuzzy c-Means (FcM) clustering of observed sample system poles.

First principles models of complex industrial processes are often derived using finite element or finite difference methods. One of the advantages of these models is that the states in the model have a clear physical interpretation. Using such models we can attempt to monitor or control selected physical quantities, even though they may not be directly measurable. Unfortunately the CPU time associated with each model evaluation of these complex models is often far too large for use in modern online monitoring or control algorithms. This paper introduces a general purpose method to approximate the computationally expensive first principles models with a quasi-Linear Parameter Varying (qLPV) model structure. The CPU time associated with the resulting qLPV models is generally considerably less than the original first principles model. The identification algorithm is such that the physical interpretation of the state vector is retained in the identified model. In contrary to other qLPV identification algorithms, the proposed algorithm extensively utilizes the availability of the original first principles model.

This paper presents results on closed-loop identification experiment design for control. The objective is to design the least costly closed-loop identification experiment such that the controller designed with the identified model stabilizes and achieves a prescribed level of H ∞ performance with the unknown true system.

For non-linear state space models, model reduction alone does not decrease the time required to compute the state update. This paper suggests methods for generating models that approximate the original reduced order models by faster equivalents. Where updating reduced order models normally requires the computation of the original large-scale model, we only compute the original large-scale model for a subset of its states. The new state of the reduced order model can then no longer be computed exactly, but has to be estimated from subset of states that have been computed using the large-scale model. It is shown that the new state of the reduced order model can be estimated accurately using spatial and temporal correlations. This acceleration method can be viewed as a partial linearization of the system equations. The methods in this paper are illustrated using a simulation example of a physical system.

Various techniques of system identification exist providing for a nominal model and uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. An amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and /spl nu/-gap uncertainty. Closed-loop performance functions based on these sets are again bounded by circles in the frequency domain, allowing for exact worst-case performance calculation and for the evaluation of the consequences of uncertainty for robust design.

This work presents a new approach for identification experiment design where the objective is to design the least costly experimental conditions such that the controller designed with the identified model stabilizes and achieves a prescribed level of H/sub /spl infin// performance with the unknown true system G/sub 0/.

In many multivariable industrial processes a subset of the available input signals is being controlled. In this paper it is analyzed in which sense the resulting partial closed-loop identification problem is actually a full closed-loop problem, or whether one can benefit from the presence of noncontrolled inputs to simplify the identification problem. The analysis focuses on the bias properties of the plant estimate when applying the direct method of prediction error identification, and the possibilities to identify (parts of) the plant model without the need of simultaneously estimating full-order noise models.

System identification in closed-loop has been of considerable interest in the last two decades. Most of the existing methods have been developed for discrete-time models. In this paper, various instrumental variable-based methods are proposed for identifying continuous-time models of systems operating in closed-loop. The accuracy of these methods is also investigated leading to the definition of the optimal IV estimator which gives minimum variance. As this needs the exact knowledge of the noise model, it cannot be used directly in practice. Several alternatives are therefore proposed to cope with this drawback and illustrated with a simulation example.

Given an expansion of a dynamical system in terms of a generalized orthonormal (Hambo) basis, the problem of realizing a state-space model of minimal McMillan degree such that its first N expansion coefficients match the given ones is addressed and solved. For the solution use is made of the properties of the Hambo operator transform theory. The resulting realization algorithms can be applied in an exact and approximative sense and can also be applied to solve a related interpolation problem.

In this paper we address the problem of computing a minimal state-space realization from partial knowledge of an expansion in terms of generalized rational orthogonal basis functions. The basis functions considered are generated by stable all-pass filters. It is shown how a minimal state-space realization can be found on the basis of complete knowledge of the expansion coefficients. Subsequently an algorithm is given that solves the partial realization problem, meaning that it computes a minimal realization on the basis of a finite number of expansion coefficients. The analysis also results in compact expressions for computing the Hambo transform underlying this basis expansion as well as its inverse. Finally it is shown how these realization problems are related to the interpolation problem of finding a rational model of minimal degree that interpolates to the derivatives of a transfer function in a given set of points outside the unit disc.

The efficiency of model predictive control can be increased by the online choice of a low number of degrees of freedom in the parametrization of the input trajectory. The complexity of the optimization is reduced by searching a solution in a lower dimensional subset of the full input space. This is done by division of the set of constraints in a set of active, inactive and possibly active constraints. A low dimensional subset of the full input space is constructed that complies with the division of the constraints implying that the optimization complexity can be decreased considerably without sacrificing too much in terms of performance.

The problem of system identification is reconsidered as a problem of deterministic approximate modeling on the basis of input-output data. In the approach presented, system identification methods are required to yield models that are well defined, in the sense that the models obtained proceed from the available data sequence and from specified users' choices, and not from implicit (statistical) assumptions about the data and the underlying process. On the basis of the system-theoretic concept of dynamical system behavior, a framework is developed in which the identification problem as considered above can be formulated properly. In this framework the different components of an identification method-model set, parameterization, and identification criterion-are defined in a fundamental and natural way. A clear distinction is made between the problems of identification and parameterization. For the popular class of equation error identification methods, it is shown that the construction of parameterizations that are identifiable by a least-squares identification criterion requires specific users' choices that have not been recognized before and that influence the optimal models obtained. >

Recently introduced methods of iterative identification and control design are directed towards the design of high performing and robust control systems. These methods show the necessity of identifying approximate models from closed loop plant experiments. In this paper a method is proposed to approximately identify normalized coprime plant factors from closed loop data. The fact that normalized plant factors are estimated gives specific advantages both from an identification and from a robust control design point of view. It will be shown that the proposed method leads to identified models that are specifically accurate around the bandwidth of the closed loop system. The identification procedure fits very naturally into the iterative identification/control design scheme as presented by Schrama (1992). >

In this paper a procedure is presented to obtain an upper bound on the modelling error for a reduced order finite impulse response (FIR) estimate of the transfer function of a linear system, using only minor a priori information. By applying a procedure similar to Bartlett's procedure of periodogram averaging to the FIR estimate, in conjunction with a periodic input signal, the statistics of the modelling error asymptotically can be obtained from the data. The modelling error consists of two parts: an averaging (probabilistic) part, due to the stochastic noise disturbance on the data, and a worst case (deterministic) part, due to the unmodelled dynamics. The latter is explicitly bounded with a hard error bound, while for the former a confidence interval can be specified asymptotically. The resulting error bounds appear to be highly realistic and, as a consequence, suitable for high performance robust control design purposes. >

Cross-covariance constraints on the noise are introduced into bounded error identification as an alternative to the standard time-domain noise constraints. The constraints are represented by a small number of linear inequalities, which can be used in parameter bounding by linear programming. An important feature of this new type of noise bound is that under fairly general conditions the feasible parameter set converges to the true parameter vector, the noise bounds need not be tight. A procedure is presented to estimate the cross-covariance bounds on the noise from data, such that asymptotically they will be correct with any prespecified probability desired. Also another noise constraint, a bound on the amplitude of the discrete Fourier transform of the noise, is introduced into time-domain parameter bounding identification. This frequency domain constraint on the noise is analysed as well. >

Classical indirect methods of closed-loop identification can be applied on the basis of different closed-loop transfer functions. Here the multivariable situation is considered and conditions are formulated under which identified approximative plant models are guaranteed to be stabilized by the present controller. Additionally it is shown in which sense the classical indirect methods are generalized by a previously introduced identification method based on the dual-Youla parametrization. For stable controllers the two methods are shown to be basically equivalent to each other.

In this paper an approach is presented to estimate a linear multivariable model on the basis of (noisy) frequency domain data via a curve fitting procedure. The multivariable model is parametrized in either a left or a right polynomial matrix fraction description and the parameters are computed by using a two-norm minimization of a multivariable output error. Additionally, input-output or element-wise based multivariable frequency weightings can be specified to tune the curve fitting error in a flexible way. The procedure is demonstrated on experimental data obtained from a 3 input 3 output wafer stepper system.

A procedure is developed to identify probabilistic frequeney response system uncertainty regions. The procedure utilizes time-domain measurement data and prior information about the system and the noise. There are no restrictions on the input signal, it may even be generated in closed loop. The system is assumed to be linear, time invariant, and a bound is assumed on the system's (generalized) pulse response parameters. The noise is assumed to be a realization of a stationary stochastic process, and independent of the input signal (in open loop operation) or an external reference signal (in closed loop operation). Frequency response confidence regions are constructed by explicitly evaluating the bias and variance errors of an instrumental variable estimate. >

Linear regression models are analysed in the least-squares identification of linear multivariable finite-dimensional models. Using the concept of system behaviour, the identification problem is formulated in a deterministic signal-oriented framework, showing clearly the distinction between problems of identification (choice of model sets) and of parametrization. In order to obtain nontrivial identified models, the identification criterion should be able to distinguish between the different models in the set. This requirement of discriminability puts restrictions on the model sets to be considered. Sets of sufficient conditions are formulated, in terms of the polynomial representations of the models, while it is noted that the identified models finally obtained are essentially dependent on the restrictions chosen. The problem discussed is shown to be closely related to the problem of constructing identifiable parameterizations for model sets described in (forward or backward) polynomial forms. >

The paper considers the problem of identifying uncertainty model sets, defined by an approximated model of the plant to be identified and a frequency domain bound on the modeling error. It is supposed that the measurements consist of time domain samples, collected in closed loop operations and corrupted by a power bounded noise. The model is supposed to be used for robust control design, whose performance is measured by a given closed loop H/sub /spl infin// norm, and the "goodness" of the model is measured by the discrepancy between the closed loop performance predicted by the model and the one actually achieved on the plant. It is shown that identifying a model minimizing this discrepancy is equivalent to finding the best approximated model of the dual Youla parametrization of the plant in a suitably weighted H/sub /spl infin// norm. Then, an optimal uncertainty model is derived for the dual Youla parametrized plant, from which an uncertainty model for the actual plant is obtained. Such an uncertainty model is finally used for designing a robust controller and evaluating the closed loop performance that can be guaranteed when the designed controller is applied to the actual plant.

For consecutive model-based control design, approximate identification of linear models should be performed on the basis of a feedback-relevant criterion, compatible with the control design. For an H/sub /spl infin//-norm based control design, a procedure is presented to estimate a possibly unstable and feedback controlled plant by using an H/sub /spl infin//-norm based feedback-relevant identification criterion. It is shown that the formulated identification problem can be handled by taking (noisy) closed loop frequency domain measurements of the plant and fitting a model of a prespecified McMillan degree, which is parametrized in a stable factorization, using a certain nonlinear constrained minimization. The procedure is illustrated by an example.

A method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain, To this end, use is made of the empirical transfer function estimate (ETFE), identified from time-domain data. As a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions, being generalizations of the classical pulse and Laguerre and Kautz types of bases. An asymptotic analysis of the estimated models is provided and conditions for consistency are formulated. Explicit and transparent bias and variance expressions are established, the latter ones also valid in a situation of undermodeling.

Radial track following of a compact disc player servo mechanism is severely exposed to periodic disturbances, induced by the eccentric rotation of the disc. The period of this disturbance is not available for measurement and varies slowly in time. Periodic disturbances can be adequately attenuated using the concept of repetitive control, provided the period is known. To deal with time varying periodic disturbances, a repetitive controller is tuned based on a simple though inaccurate physical model of the time varying character of the period. The model is tuned based on an estimate of the period, obtained through recursive identification. Experimental results show that the proposed scheme enables a significant improvement of the tracking accuracy of the radial servo mechanism.

The powerful standard representation for uncertainty descriptions in a basic perturbation model based on a standard plant representation can be used to attain necessary and sufficient conditions for stability robustness within various uncertainty descriptions. In this paper these results are employed to formulate necessary and sufficient conditions for stability robustness of several uncertainty sets based on unstructured additive coprime factor uncertainty, gap-metric uncertainty as well as /spl Lambda/-gap uncertainty.

A general transform theory is presented that underlies expansions of stable discrete-time transfer functions in terms of rational orthonormal bases. The types of bases considered are generated by cascade connections of stable all-pass functions. If the all-pass sections in such a network are all equal, this gives rise to the Hambo basis construction. In the paper a more general construction is studied in which the all-pass functions are allowed to be different, in terms of choice and number of poles that are incorporated in the all-pass functions. It is shown that many of the interesting properties of the so-called Hambo transform that underlies the Hambo basis expansion carry over to the general case. Especially the expressions for the computation of the Hambo transform on the basis of state-space expressions can be extended to the general basis case. This insight can for instance be applied for the derivation of a recursive algorithm for the computation of the expansion coefficients, which are then obtained as the impulse response coefficients of a linear time-varying system.

The identification of linear models that are particularly suitable for serving as a basis for (robust) model-based control design has recently attracted considerable attention. Both the system identification community and the control community have spent considerable efforts in developing a coherent approach to the problem. Typical problems that have to be dealt with consider questions of optimal experiment design, feedback-relevant system approximations and control-relevant model uncertainty specifications. Research into these problems has delivered several attempts for bridging the gap between identification and control theory. In this lecture these developments are highlighted, directing particular attention to the identification of control-relevant approximate models, the use of closed-loop experimental data for identification, the quantification of model uncertainty, and the use of identification criteria that are motivated by control performance cost functions.

A variance reduction scheme is presented for non-parametric transfer function estimators based on the use of wavelets as an alternative to the traditional spectral windowing. The latter can be generalized into a variance reduction method based on thresholding (omitting or altering) the coefficients of an orthogonal series expansion of the estimator to be smoothed. The choice of threshold level is crucial, distinguishing between coefficients related predominantly to estimation errors and those associated with the underlying true function. The standard wavelet threshold operation with a constant or level-dependent threshold can not be applied to wavelet coefficients of spectral density functions. The nonstationarity in the statistical properties of these estimators reveals itself in the wavelet domain as significant peaks. An efficient threshold level should follow the standard deviation of each wavelet coefficient. New exact expressions of the standard deviation are presented. An estimator based on these expressions is proved to provide an appropriate threshold level.

This paper proposes a new robust control design procedure based on a model and an uncertainty region deduced from classical PE identification. The key step in the procedure is a quality assessment procedure for the pair “model-uncertainty region” taking into account the prescribed performance level.

This paper discusses the approximation and feedback relevant parametric identification of a positioning mechanism present in a wafer stepper. The positioning mechanism in a wafer stepper is used in chip manufacturing processes for accurate positioning of the silicon wafer on which the chips are to be produced. The accurate positioning requires a robust and high-performance feedback controller that enables a fast throughput of silicon wafers. A control relevant set of multivariable finite dimensional linear time invariant discrete-time models is formulated and estimated on the basis of closed-loop experiments. The set of models is shown to be suitable for model-based robust control design of the positioning mechanism. This is illustrated by a successful design and implementation of a robust controller.

This paper considers the approximation of linear systems by means of orthonormal basis functions, which are generated by stable all-pass functions. These basis functions induce the so called Eambo transform, which transforms scalar systems into square systems of i/o dimension equal to the order of the all-pass function considered. We will consider the construction of the Markov parameters of the system representation in the transform domain and show how these can be used to realize minimal state space representations for the exact and partial knowledge case. Additionally a projection mechanism is presented to allow inverse transformation of any sequence of Markov parameters in the transform domain. This mechanism is illustrated with an example.

Multivariable system identification of a model IV fluidized catalytic cracking unit is performed using a linear time invariant model parametrization based on orthonormal basis functions. This model structure is a linear regression structure which results in a simple convex optimization problem for least squares prediction error identification. Unknown initial conditions are estimated simultaneously with the system dynamics to account for the slow drift of the measured output from the given initial condition to a stationary working point. The model accuracy for low frequencies is improved by a steady-state constraint on the estimated model and incorporation of prior knowledge of the large time constants in the model structure. The model accuracy is furthermore improved by an iteration over identification of a high order model and model reduction. First a high order model is estimated using an orthonormal basis. This model is reduced and used to generate a new orthonormal basis which is used in the following iteration step for high order estimation. With the approach followed accurate models over a large frequency range are estimated with only a limited amount of data.

This paper discusses the control relevant parametric identification of a servo system present in a compact disc player. In this application an approximate closed loop identification problem is solved in order to come up with a linear multivariable discrete time model, suitable for control design. This identification problem is handled by a recently introduced two stage method. It yields an explicit and tunable expression for the bias distribution of the model being estimated, clearly showing the dynamics or the closed loop system in the (asymptotic) approximation criterion. This result is exploited to identify the model in a control relevant way by additional data filtering. The recently introduced method by Vries-Van den Hof (1993) for model uncertainty quantification is used to construct an upper bound for the corresponding model error. >

The Ho-Kalman algorithm creates a minimum realization of a system, when given a series of deterministic Markov parameters. However, when such a ‘truncated’ series of Markov parameters has been disturbed with noise, an approximating Hankel matrix has to be constructed for applying the realization algorithm. This approximating Hankel matrix has either the improper rank, or it lacks the Hankel structure. Furthermore the Markov parameters are not processed with a constant weighting factor, which implies that the noise filtering is inadequate. In this paper we propose to use an alternative matrix: the Page matrix. It is shown that this method is better suited for handling the noisy Markov parameters. This holds with respect to three aspects: order testing, noise filtering and realization. Even in the deterministic case, the Page matrix offers the advantage of a considerable reduction in computation.

This paper deals with the minimal partial realization problem for a multivariable power series of finite length. Specifically, the problem of the uniqueness of the extension sequence is considered and a fundamental criterion is given which is necessary and sufficient for the existence of a unique extension. A specific class of formal power series is introduced (finite generic sequences) for which the criterion mentioned above takes a very simple form. This class is of interest when considering the availability of a sequence of noise disturbed Markov parameters of a multivariable system. Finally the approximate partial realization problem is defined.

The problem of system identification is considered in terms of (approximate) modelling of a given sequence of measurement data. Given a data sequence of a multivariable process, there arc two aspects that play a central role in determining the properties of the identified models: the set of models that is taken into account, and the identification criterion that selects the identified models from this set. When parametrizing a set of models ℳ very often use is made of a notion of model equivalence in order to arrive at a uniquely identifiable model set ℳ* ⊂ ℳ It is undesirable that the dynamical properties of the identified model are dependent on the specific choice of ℳ*, i.e. the specific parametrization chosen; consequently it is shown that the applied model equivalence, which conventionally refers to input–output equivalence, has to be in accordance with the identification criterion. Therefore a criterion based model equivalence is introduced which for equation error identification methods leads to a non conventional set of canonical forms.

This paper presents experiments for finding a model of the cognitive behaviour of the human operator in some well-defined task. The task chosen is inventory control. Results of the modelling are compared with the performance of the human operator and these are discussed in the context of the optimal controller for the inventory control.

This paper presents suboptimal solutions to the problem of Approximate Partial Realization: given a multivariable formal power series of finite length, a state space description has to be constructed with limited dimension, in such a way that it approximates the available matrix sequence with respect to some specific criterion. The methods based on the Ho-Kalman algorithm and employing a Hankel matrix or an alternative to it, a Page matrix, fail theoretically but appear to be practically quite useful. Results of the simulations allow us to determine in which cases the Hankel or Page matrix approach would be more appropriate.

In this paper some aspects of the partial realization problem for multivariable formal power series are considered. The dynamic structure of the systems underlying such series is analysed on the basis of the results of Kalman, and a canonical realization algorithm is described based on this concept of structure. Special attention is paid to the aspect of uniqueness of the realization; necessary and sufficient conditions are given for the uniqueness of the extension sequence. A specific class of power series is defined for which a very simple uniqueness criterion is derived. Generally, noise-disturbed data sequences belong to this specific class; it is shown that the unique partial realization for such a noise-disturbed power series may serve as an intermediate step to arrive at an approximate realization of limited order.